The sandwich line graph

Denis Cornaz, Philippe Meurdesoif
2010 Electronic Notes in Discrete Mathematics  
We observe that ω(G) + χ(S( G)) = n = ω(S( G)) + χ(G) for any graph G with n vertices, where G is any acyclic orientation of G and where S( G) is the (complement of the) auxiliary line graph introduced in [1]. (Where as usual, ω and χ denote the clique number and the chromatic number.) It follows that, for any graph parameter β(G) sandwiched between ω(G) and χ(G), then Φ β ( G) := n−β(S( G)) is sandwiched between ω(G) and χ(G) too. Numerical experiments show that Φ ϑ is closer to χ than ϑ, where ϑ is Lovász theta function.
doi:10.1016/j.endm.2010.05.121 fatcat:pbbrwywli5dajmrbkiroi3pisi